About Appearance of the Irreversibility

The inevitability of arising in equations of kinetics and hydrodynamics irreversibility not contained in original equations of classic mechanics is substantiated. It is established that transfer of information about the direction of system evolution from initial conditions to resulting equations is the consequence of losing information about the position of an individual particle in space, which takes place at roughening description. It is shown that the roughening with respect to impact parameters of colliding particles is responsible for appearance of the irreversibility in resulting equations. Direct equations of kinetics and hydrodynamics are the result of roughening distribution functions with respect to impact parameters of particles, which have not yet reached the domain of their interaction. The direct equations are valid for the progressive direction of timing on the time axis pointing from the past to the future. Reverse equations of kinetics and hydrodynamics are the result of roughening distribution functions with respect to impact parameters of particles, which have already left the domain of their interaction. The reverse equations are valid for the progressive direction of timing on the time axis pointing from the future to the past.

Multimoment Hydrodynamics in Problem on Flow around a Sphere: Entropy Interpretation of the Appearance and Development of Instability

Multimoment hydrodynamics equations are applied to investigate the phenomena of appearance and development of instability in problem on a flow around a solid sphere at rest. The simplest solution to the multimoment hydrodynamics equations coincides with the Stokes solution to the classic hydrodynamics equations in the limit of small Reynolds number values, . Solution ?to the multimoment hydrodynamics equations reproduces recirculating zone in the wake behind the sphere having the form of an axisymmetric toroidal vortex ring. The ?solution remains stable while the entropy production in the system exceeds the entropy outflow through the surface confining the system. The passage of the first critical value ?is accompanied by the ?solution stability loss. The ?solution, when loses its stability, reproduces periodic pulsations of the periphery of the recirculating zone in the wake behind the sphere. The ?and ?solutions to the multimoment hydrodynamics equations interpret a vortex shedding. After the second critical value ?is reached, the ?solution at the periphery of the recirculating zone and in the far wake is replaced by the ?solution. In accordance with the ?solution, the periphery of the recirculating zone periodically detached from the core and moves downstream in the form of a vortex ring. After the attainment of the third critical value , the ?solution at the periphery of the recirculating zone and in the far wake is replaced by the ?solution. In accordance with the ?solution, vortex rings penetrate into each other and form the continuous vortex sheet in the wake behind the sphere. The replacement of one unstable flow regime by another unstable regime is governed the tendency of the system to discover the fastest path to depart from the state of statistical equilibrium. Having lost the stability, the system does not reach a new stable position. Such a scenario differs from the ideas of classic hydrodynamics, which interprets the development of instability in terms of bifurcations from one stable state to another stable state. Solutions to the multimoment hydrodynamics equations indicate the direction of instability development, which qualitatively reproduces the experimental data in a wide range of Re values. The problems encountered by classic hydrodynamics when interpreting the observed instability development process are solved on the way toward an increase in the number of principle hydrodynamic values.

Kinetic Foundation for the Multimoment Hydrodynamics Equations

The equations for the pair distribution functions are derived directly from the second equation of the Bogolyubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy. The derivation is fulfilled within the frameworks of the multiscale method. The equations for the pair distribution functions are the kinetic foundation for the multimoment hydrodynamics equations. Solutions to the equations for the pair distribution functions predetermine the possibility of constructing the hydrodynamics equations with an arbitrary number of principle hydrodynamic values specified beforehand. The tendency to increase the number of principal hydrodynamic values is caused by the necessity of interpreting the behavior of the system after the loss of stability. Solutions to the classic hydrodynamics equations constructed for only three principle hydrodynamic values are unable to predict the direction of instability evolution. Solutions to the multimoment hydrodynamics equations are capable of reproducing correctly the phenomenon of emergence and development of instability.